Give the spaces of connections and sections the structure of Banach
manifolds by taking completions with respect to Sobolev norms \(1,p\) 
for 1-forms, and \(0,p\)  for 0 and 2-forms. For \(p>2\) , the map
\(\mathcal {F}_ {A,u}\)  is a smooth map of Banach spaces.
\(\mathcal {F}_{A,u} : \Omega ^1(\Sigma ,P(\mathfrak {g}))_{1,p} \oplus \Omega ^0(\Sigma ,u^*T^{\operatorname{vert}} P(X))_{1,p} \\ \rightarrow (\Omega ^0 \oplus \Omega ^2)(\Sigma ,P(\mathfrak {g}))_{0,p} \oplus \Omega ^{0,1}(\Sigma ,u^*T^{\operatorname{vert}} P(X))_{0,p} \) 
equivariant for the action of the group \(\mathcal {G}(P)_{2,p}\)  of gauge
transformations of class \(2,p\) . Suppose that \((A,u)\)  is regular and
stable. By the implicit function theorem, there is a local
homeomorphism
\(  \operatorname{ker}(\tilde{D}_{A,u}) \rightarrow \left\lbrace  \begin{array}{c} F_{A+ a} +  \operatorname{Vol}_\Sigma (\exp _u(\xi ))^* P(\Phi ) = 0 \\ \overline{\partial }_{A + a}(\exp _u(\xi )) = 0 \\ \ {\operatorname{d}}_{A,u}^*(a,\xi ) = 0 \end{array} \right\rbrace .\) 
This gives rise to a family \((A_S,u_S) \rightarrow S\)  over a neighborhood \(S\) 
of 0 in \( \operatorname{ker}(\tilde{D}_{A,u})\) . By [1]},
\((A_S,u_S)\)  is a smooth family, assuming \((A,u)\)  is smooth. Given any
other family \((A_{S^{\prime }}^{\prime },u_{S^{\prime }}^{\prime }) \rightarrow S^{\prime }\)  of stable vortices with
\((A^{\prime }_0,u_0^{\prime }) = (A,u)\) , the implicit function theorem provides a smooth
map \(S^{\prime } \rightarrow S\)  so that \((A^{\prime }_{S^{\prime }},u^{\prime }_{S^{\prime }})\)  is obtained from \((A,u)\)  by
pull-back. The first property of the universal deformation is a
consequence of the slice condition; the second property follows from
the fact that the projection \( \operatorname{ker}(\tilde{D}_{A,u}) \rightarrow  \operatorname{ker}(\tilde{D}_{A_s,u_s})\)  is an isomorphism for sufficiently small \(s\) .
